Download Normal Probability Plots

TE C H N OL OG Y
E X T EN S I O N
Normal Probability Plots
If it is not known whether the underlying population is normally distributed,
you can use a graphing calculator or software to construct a normal probability
plot of the sample data. A normal probability plot graphs the data according to
the probabilities you would expect if the data are normal, using z-scores. If the
plot is approximately linear (a straight line), the underlying population can be
assumed to be normally distributed.
Using a Graphing Calculator
A toy tricycle comes with this label: “Easy-To-Assemble. An adult can complete
this assembly in 20 min or less.” Thirty-six adults were asked to complete the
assembly of a tricycle, and record their times. Here are the results:
16 10 20 22 19 14 30 22 12 24 28 11 17 13 18 19 17 21
29 22 16 28 21 15 26 23 24 20 8
17 21 32 18 25 22 20
1. Using a graphing calculator, enter these data in L1. Find the mean and
standard deviation of the data.
2. Make a normal probability plot of the data. Using STAT PLOT, select 1:Plot1,
and the settings shown below.
Based on the plot, are assembly times normally distributed?
3. a) What is the probability that an adult can complete this assembly in
20 min or less?
b) What proportion of adults should complete this assembly within 15 to
30 min?
Using Fathom™
4. a) Open FathomTM, and open a new document if necessary. Drag a new
collection box to the workspace. Rename the collection Assembly Times,
and create 36 new cases.
b) Drag a new case table to the workspace. Name the first column Times,
the second column zTimes, and the third column Quantiles.
442
MHR • The Normal Distribution
c) i) Enter the time data in the first column. Sort it in ascending order.
ii) Edit the formula in the second column to zScore(Times). This will
calculate the z-scores for the data.
iii) Edit the formula in the third column to
normalQuantile((uniqueRank(Times) – 0.5)/36, 0, 1).
This formula will calculate the z-scores of the quantiles
corresponding to each entry in the Times column. The uniqueRank( )
function returns the “row number” of the sorted data. Note that most
of the quantile z-scores in the screen below are different from the zscores for the corresponding data.
d) Drag a new graph to the workspace. Drag the Times title to the
horizontal axis, and the Quantiles title to the vertical axis to generate a
normal probability plot. Calculate the linear correlation coefficient for
Times and Quantiles and comment on how near to linear this graph is.
Are the data normally distributed?
Technology Extension: Normal Probability Plots • MHR
443
e) Double click on the collection to open the inspector. Choose the Measures
tab. Create four measures: Mean, StdDev, P20orLess, and P15to30. Use the
mean, standard deviation, and normalCumulative functions to calculate the
mean, the standard deviation, and the answers to question 3.
5. For each of the questions 1 to 6, 8, and 12 of section 8.3, pages 439 to 441,
use a normal probability plot to determine how close to a normal
distribution each data set is.
6. Let x1, x2, … xn be a set of data, ranked in increasing order so that
x1 ≤ x2 ≤ … ≤ xn.
(i – 0.5)
For i = 1, 2,…, n, define the quantile zi by P(Z < zi ) = ,
n
where Z is a standard normal distribution (mean 0, standard deviation 1).
a) For a data set of your choice, plot a graph of zi against xi. Remember to
sort the x-values into increasing order. Use the invNorm( function on your
graphing calculator, or the table of Areas Under the Normal Distribution
Curve on pages 606 and 607, to calculate the z-values. Notice that these
quantile z-values are different from the z-scores in earlier sections.
b) Compare this graph with the normal probability plot for the data set.
Explain your findings.
c) Explain why, if the data are normally distributed, a graph of zi against xi
should be close to a straight line.
444
MHR • The Normal Distribution